Math / Geometry

QuestionGiven two points $A(8,2)$ and $B(-1,-4)$ , the line segment $A B$ cuts the $x$ -axis at $P(x, 0)$ , find the value of $x$ .

Studdy Solution

STEP 1

What is this asking? We need to find where the line connecting two points crosses the horizontal axis. Watch out! Don't mix up the $x$ and $y$ coordinates!
Also, remember that the $y$ -coordinate is zero where a line crosses the horizontal axis.

STEP 2

1. Find the slope
2. Find the equation of the line
3. Find the x-intercept

STEP 3

Let's start by finding the slope of the line segment $AB$ .
Remember, the slope is the rise over run, or how much the $y$ -value changes for every change in the $x$ -value.

STEP 4

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Let's plug in our points $A(8, 2)$ and $B(-1, -4)$ .
We'll call $(8, 2)$ point 1, so $x_1 = 8$ and $y_1 = 2$ .
Then $(-1, -4)$ is point 2, so $x_2 = -1$ and $y_2 = -4$ .

STEP 5

Plugging those values into our formula gives us $m = \frac{-4 - 2}{-1 - 8} = \frac{-6}{-9}$ .
Now, we can simplify by dividing both the numerator and the denominator by $-3$ , which is like multiplying by one in the form of $\frac{-3}{-3}$ .
This gives us $m = \frac{-6}{-9} \cdot \frac{-3}{-3} = \frac{2}{3}$ .
So, our slope is $\frac{2}{3}$ .

STEP 6

Great! Now we have the slope.
We can use the point-slope form of a line, which is $y - y_1 = m(x - x_1)$ .
Let's use point $A(8, 2)$ and our slope $m = \frac{2}{3}$ .

STEP 7

Substituting, we get $y - 2 = \frac{2}{3}(x - 8)$ .
Let's distribute that $\frac{2}{3}$ to get $y - 2 = \frac{2}{3}x - \frac{16}{3}$ .

STEP 8

Now, let's add $2$ to both sides to isolate $y$ .
Remember, we can write $2$ as $\frac{6}{3}$ so that we have a common denominator.
This gives us $y = \frac{2}{3}x - \frac{16}{3} + \frac{6}{3}$ , which simplifies to $y = \frac{2}{3}x - \frac{10}{3}$ .
This is the equation of our line!

STEP 9

The $x$ -intercept is where the line crosses the $x$ -axis, which means the $y$ -coordinate is zero.
So, we can substitute $y = 0$ into our equation: $0 = \frac{2}{3}x - \frac{10}{3}$ .

STEP 10

Now, let's add $\frac{10}{3}$ to both sides to get $\frac{10}{3} = \frac{2}{3}x$ .

STEP 11

To solve for $x$ , we can multiply both sides by $\frac{3}{2}$ .
This is like multiplying by one in the form of $\frac{3/2}{3/2}$ , which isolates $x$ .
We get $\frac{3}{2} \cdot \frac{10}{3} = \frac{3}{2} \cdot \frac{2}{3}x$ .

STEP 12

Simplifying gives us $\frac{30}{6} = x$ , which reduces to $x = 5$ .
So, the $x$ -intercept is $5$ !

STEP 13

The line segment $AB$ intersects the $x$ -axis at the point $P(5, 0)$ .

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